Conification of Kahler and Hyper-Kahler Manifolds

Alekseevsky, DV, Cortes, V and Mohaupt, T ORCID: 0000-0002-6864-4086 (2013) Conification of Kahler and Hyper-Kahler Manifolds. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 324 (2). pp. 637-655.

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Abstract

Given a K\"ahler manifold $M$ endowed with a Hamiltonian Killing vector field $Z$, we construct a conical K\"ahler manifold $\hat{M}$ such that $M$ is recovered as a K\"ahler quotient of $\hat{M}$. Similarly, given a hyper-K\"ahler manifold $(M,g,J_1,J_2,J_3)$ endowed with a Killing vector field $Z$, Hamiltonian with respect to the K\"ahler form of $J_1$ and satisfying $\mathcal{L}_ZJ_2= -2J_3$, we construct a hyper-K\"ahler cone $\hat{M}$ such that $M$ is a certain hyper-K\"ahler quotient of $\hat{M}$. In this way, we recover a theorem by Haydys. Our work is motivated by the problem of relating the supergravity c-map to the rigid c-map. We show that any hyper-K\"ahler manifold in the image of the c-map admits a Killing vector field with the above properties. Therefore, it gives rise to a hyper-K\"ahler cone, which in turn defines a quaternionic K\"ahler manifold. Our results for the signature of the metric and the sign of the scalar curvature are consistent with what we know about the supergravity c-map.

Item Type:	Article
Additional Information:	conjecture replaced by reference
Uncontrolled Keywords:	math.DG, math.DG, hep-th, 53C26
Subjects:	?? QA ?? ?? QC ??
Depositing User:	Symplectic Admin
Date Deposited:	27 Oct 2015 10:27
Last Modified:	16 Dec 2022 08:20
DOI:	10.1007/s00220-013-1812-0
Related URLs:	Author Publisher
URI:	https://livrepository.liverpool.ac.uk/id/eprint/2033939